1.

Three point charges + q each are kept at the vertices of an equilateraltriangleof side l. Determinethe magnitude and sign of charge to be kept at the centroid so that charges at the vertices remain in equillibrium.

Answer»

`Q = (2q)/(sqrt(3))`
`Q = (q)/(sqrt(2))`
`Q = (3q)/(sqrt(3))`
`Q = (q)/(sqrt(3))`

Solution :In fig. 1(a).21,let Q be the chargerequired to be PLACED at G. Now.
`vec(F_(1))` = force at A due to charge at B
`= (1)/(4pi in_(0)) (q^(2))/(L^(2))`, along `vec(BA)`
`vec(F_(2))` = forceat A due to charge at C
`= ((1)/(4pi in_(0)) (q^(2))/(l^(2)))`, along `vec(CA)`.
`vec(F_(1)) + vec(F_(2)) = sqrt(F_(1)^(2) + F_(2)^(2) + 2F_(2)F_(2) cos60^(0))`
`= F_(1) sqrt(3) = (sqrt(3)q^(2))/(4pi in_(0) l^(2))` along GA.
Force at A due to chargeQ at the CENTROID G
`= (1)/(4pi in_(0)) (Qq)/(AG^(2)) = (Qq)/(4pi in_(0) (l// sqrt(3))^(2)) = (3Qq)/(4pi in_(0) l^(2))`
This mustbe EQUALAND opposite to `(vec(F_(1)) + vec(F_(2)))`
`:. (3Qq)/(4pi in_(0) l^(2)) =(- sqrt(3) q^(2))/(4pi in_(0) l^(2))`
`Q = (q)/(sqrt(3))`


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